Exact spectrum of the dilute oriented loop model: general branching rules into dense SU(n+1) modules and mixed-parity scaling dimensions

The dilute oriented loop model of Vernier–Jacobsen–Saleur (J. Phys. A 49 (2016) 064002) places oriented loops on a square lattice whose edges carry a fixed alternating orientation, so strands never go straight; the model has U( $n$ ) symmetry, a critical line in the dilute O( $2n$ ) universality class for vertex weight $\tau<2$ , and a special point $(K,\tau)=(\tfrac12,2)$ coinciding with the dense loop model of fugacity $n+1$ . Two parts of its exact finite-size spectrum were left open: the branching rules by which the dilute-oriented standard modules recombine into dense SU( $n+1$ ) modules at the special point were exhibited only for $L=2$ , and along the $n=1$ critical line — where the model maps onto the six-vertex model — the scaling dimensions of sectors mixing even/odd through-line parity labels in a non-alternating way were computed only numerically, with no analytical formula or convincing conjecture. This problem asks for the closed analytical characterization of both: the general- $L$ branching rules and the mixed-parity watermelon (multi-leg) scaling dimensions, together with the identification of the corresponding continuum operators.

Background

A loop model is a lattice statistical model whose configurations are non-intersecting closed curves, each loop carrying a fugacity (weight) $n$ ; for a range of fugacities such models have critical points described by two-dimensional conformal field theory. In the standard dilute O( $n$ ) model some lattice edges are left empty, giving a dilute critical phase. Vernier, Jacobsen and Saleur (2016) introduced a dilute oriented loop model on the square lattice in which the lattice edges carry a fixed, alternating orientation that loop strands must follow: strands are never allowed to go straight at a vertex, and the resulting symmetry is enhanced from O( $n$ ) to U( $n$ ). The model has two parameters: a monomer weight $K$ controlling dilution and a vertex weight $\tau$ . Its transfer matrix acts on link states — the connectivity states of the $2L$ lattice edges crossed by the transfer direction — and decomposes into standard modules, that is, sectors labelled by the number $j$ of through-lines (line segments crossing the transfer direction) together with, for the oriented model, a parity label e or o attached to each through-line according to the parity of the lattice edge it occupies (edges within a row alternate e-o-e-o from left to right, e denoting even and o odd positions); these parity labels are individually conserved by the transfer matrix. At a given size $2L$ , the admissible parity label sequences are those realizable as sub-sequences of the basic pattern eoeo… . The paper computed the dimensions of these modules combinatorially and algebraically (the transfer matrix generates a 'dilute oriented' version of the Temperley–Lieb algebra, the diagram algebra of non-crossing connectivities originally introduced for the six-vertex and Potts models). It established the phase diagram: for generic $n\le 1$ a critical line for $\tau<2$ in the universality class of the dilute O( $2n$ ) model (i.e. loops of fugacity $2n$ ), separating a massive dilute phase from a dense O( $n$ ) phase, and terminating at the special point $(K,\tau)=(\tfrac12,2)$ , where the transfer matrix can be written in terms of the ordinary Temperley–Lieb algebra with loop fugacity $n+1$ , so the model coincides with the dense loop model whose continuum limit is the SU( $n+1$ ) Wess–Zumino–Witten (WZW) conformal field theory, the exactly solved theory governed by affine-Lie-algebra current symmetry. Exactly at this point the spectra of the two formulations must recombine: eigenvalues in the dilute-oriented modules become degenerate with eigenvalues of the dense modules with appropriate multiplicities; the authors verified this explicitly only at size $L=2$ — separately for open (free) boundary conditions and for periodic boundary conditions, where parity labels differing by a cyclic permutation are identified and each sector of $j$ through-lines carries a seam phase $\varphi=2\pi m/j$ ( $m=0,\ldots,j-1$ ) — and left the branching rules for general $L$ as, in their words, an interesting algebraic exercise beyond the scope of the paper. Separately, for $n=1$ the model maps exactly onto the six-vertex model along the critical line, with anisotropy $\Delta=1-\tau$ ; the critical regime covers $\tau\in(0,2]$ (i.e. $\Delta\in[-1,1)$ ), its endpoint $\tau=2$ ( $\Delta=-1$ , $\bar g=1$ ) being the SU(2) point, and its $c=1$ Coulomb-gas conformal field theory (a compactified free boson, with central charge $c=1$ ) gives scaling dimensions $x_{E,M}=E^{2}/(2\bar{g})+\bar{g}M^{2}/2$ , where $E$ and $M$ are the electric and magnetic charges and $\Delta=\cos(\pi\bar{g})$ . The watermelon, or multi-leg, scaling exponents are the scaling dimensions of operators carrying $\ell$ through-lines (loop strands) through a point. Pure-parity sectors correspond to $M=\pm\ell$ , $E=0$ , giving $x=\bar{g}\,\ell^{2}/2$ , and purely alternating parity labels give constants $x=\ell^{2}/2$ ; but all sectors mixing e and o labels are absent from the local six-vertex spectrum — their operators are non-local in the six-vertex formulation — and for those mixing e and o in a non-purely-alternating way (for example the sequences eoo, eoee, eoeoo) the exponents, known only from periodic transfer-matrix diagonalization up to $L=16$ , vary along the line with no analytical formula and no convincing conjecture.

Problem Statement

The problem is to complete the exact analytical characterization of the transfer-matrix spectrum of the dilute oriented loop model on the square lattice (loop fugacity $n$ , monomer weight $K$ , vertex weight $\tau$ , transfer matrix acting on $2L$ sites), which the 2016 analysis of Vernier–Jacobsen–Saleur established only through finite-size examples and numerics. Two linked results are required, characterizing the two ends of the same critical line:

Branching rules at the SU( $n+1$ ) point. At the special point $(K, \tau) = (\tfrac{1}{2}, 2)$ the transfer matrix can be written in terms of both the dilute oriented Temperley–Lieb algebra with parameter $n$ and the ordinary (dense) Temperley–Lieb algebra with parameter $n+1$ , so the eigenvalues in the dilute-oriented standard modules — labelled by the number of through-lines and their sequence of even/odd parity labels — must recombine into degeneracy classes matching the dense standard modules. Derive these branching rules for arbitrary system size $L$ : an explicit multiplicity formula or algorithm giving, for each $L$ and each admissible parity labelling (at size $2L$ , a labelling realizable as a sub-sequence of the basic pattern eoeo…), how the dilute-oriented modules map into the dense modules of the SU( $n+1$ ) model, generalizing the $L = 2$ decompositions exhibited in the source for each of the two boundary geometries it treats: open (free) boundary conditions, and periodic boundary conditions (where parity labels differing by a cyclic permutation are identified and each sector of $j$ through-lines carries a seam phase $\varphi=2\pi m/j$ , $m=0,\ldots,j-1$ ). The rules must be consistent with the combinatorial dimension sum rules of the standard modules and reproduce the transfer-matrix eigenvalue degeneracies.
Mixed-parity scaling dimensions at $n = 1$ . For $n = 1$ the model maps onto the critical six-vertex line, whose critical regime covers $\tau \in (0,2]$ (anisotropy $\Delta = 1-\tau \in [-1,1)$ ), with Coulomb-gas parameter $\bar{g}$ defined by $\Delta = \cos(\pi \bar{g})$ ; the endpoint $\tau = 2$ ( $\Delta = -1$ ) is the SU(2) point, where $\bar{g} = 1$ . Derive explicit analytical formulas for the bulk (periodic-boundary) watermelon (multi-leg) scaling exponents — the scaling dimensions of the operators carrying the corresponding through-lines — of the sectors whose parity labels mix e and o in a non-purely-alternating way (e.g., $x_{\mathrm{eoo}}$ , $x_{\mathrm{eoee}}$ , $x_{\mathrm{eoeoo}}$ ), for which the source has neither a formula nor a convincing conjecture. (All sectors with mixed parity labels — including the purely alternating ones — are absent from the local six-vertex spectrum, their operators being non-local in the six-vertex formulation; the continuum-operator identification must account for this non-locality.) Identify the corresponding continuum operators. The formulas must reproduce the pure-parity exponents $x_{\mathrm{oo\ldots o}(\ell)} = \bar{g}\,\ell^2/2$ established in the source with strong numerical support, the SU(2)-point values at $\tau = 2$ , and the available finite-size data along the line; for the purely alternating labels the source conjectures $x_{\mathrm{eoeo}\ldots(\ell\ \mathrm{pairs})} = \ell^2/2$ on the basis of the $\ell = 1,2$ numerics ( $x_{\mathrm{eo}}=x_{\mathrm{oe}}=\tfrac12$ , $x_{\mathrm{eoeo}}=x_{\mathrm{oeoe}}=2$ ), so the formulas must be consistent with those numerics, but the conjecture itself is not an established benchmark that they are required to reproduce at larger $\ell$ .

Deliverables 1 and 2 jointly complete the exact operator content of the model's critical line — deliverable 1 fixing the module-level recombination at the dense endpoint, deliverable 2 the conformal dimensions of all parity-labelled sectors at $n = 1$ — and an answer is complete when the branching rules of deliverable 1 hold for arbitrary $L$ at the special point $(K,\tau)=(\tfrac12,2)$ and the formulas of deliverable 2 hold for all $\tau$ on the $n=1$ critical line ( $0<\tau\le 2$ ), not merely at inspected finite sizes.

References

Eric Vernier, Jesper Lykke Jacobsen, Hubert Saleur, "Dilute oriented loop models", J. Phys. A: Math. Theor. 49 (2016) 064002. DOI: 10.1088/1751-8113/49/6/064002. arXiv:1509.07768. URL: https://arxiv.org/abs/1509.07768
Paul Roux, Jesper Lykke Jacobsen, Sylvain Ribault, Hubert Saleur, "Critical spin chains and loop models with PSU(n) symmetry", SciPost Phys. 18 (2025) 033. DOI: 10.21468/SciPostPhys.18.1.033. arXiv:2404.01935. URL: https://arxiv.org/abs/2404.01935
Jesper Lykke Jacobsen, Sylvain Ribault, Hubert Saleur, "Spaces of states of the two-dimensional O(n) and Potts models", SciPost Phys. 14 (2023) 092. DOI: 10.21468/SciPostPhys.14.5.092. arXiv:2208.14298. URL: https://arxiv.org/abs/2208.14298
Adam Nahum, "Universality class of the two-dimensional polymer collapse transition", Phys. Rev. E 93 (2016) 052502. DOI: 10.1103/PhysRevE.93.052502. arXiv:1510.09223. URL: https://arxiv.org/abs/1510.09223
Éric Vernier, Jesper Lykke Jacobsen, Hubert Saleur, "A new look at the collapse of two-dimensional polymers", J. Stat. Mech. (2015) P09001. DOI: 10.1088/1742-5468/2015/09/P09001. arXiv:1505.07007. URL: https://arxiv.org/abs/1505.07007
Romain Couvreur, Eric Vernier, Jesper Lykke Jacobsen, Hubert Saleur, "On truncations of the Chalker–Coddington model", Nucl. Phys. B 940 (2019) 196. DOI: 10.1016/j.nuclphysb.2019.02.015. arXiv:1809.07429. URL: https://arxiv.org/abs/1809.07429